Efficiently adaptive double pyramidal coding

ABSTRACT

In accordance with an embodiment, a method of encoding includes generating for each transform point a double difference coefficient (comprising the difference between a modeled difference coefficient and a raw difference coefficient) and encoding as an adaptive difference coefficient for each transform point either the double difference coefficient or the raw difference coefficient. Whether the double difference coefficient or the raw difference coefficient is selected to be the adaptive difference coefficient depends on which one provides more efficient coding. A method of decoding includes receiving the adaptive difference coefficients from the encoder, applying the same modeling and transform as the encoder to generate the modeled difference coefficients, generating corrective difference coefficients (from the adaptive difference coefficients and the modeled coefficients), and inverse transformation using the corrective difference coefficients. A system may include an encoder implementing the method of encoding and a decoder implementing the method of decoding.

CROSS-REFERENCES TO RELATED APPLICATIONS

[0001] This patent application claims priority from U.S. ProvisionalPatent Application No. 60/257,845, filed Dec. 19, 2000 and entitled“Double Pyramid,” the disclosure of which is hereby incorporated byreference. This patent application is related to commonly-owned andco-pending U.S. patent application Ser. No. 10/______,______, (attorneydocket #10006.000610), filed Dec. 19, 2001 and entitled “AdaptiveTransforms,” inventors Adityo Prakash, Edward Ratner, and Dmitri Antsos,the disclosure of which is hereby incorporated by reference.

BACKGROUND OF THE INVENTION

[0002] 1. Field of the Invention

[0003] The present invention relates in general to communications. Moreparticularly, the invention relates to the transmission ofmultidimensional signals, such as video signals.

[0004] 2. Description of the Background Art

[0005] Motivation

[0006] The transmission of large amounts of data across a largedecentralized network, such as the Internet, is an open problem, Motionpicture data, i. e., video data, presents a particularly vexing problem,as the data tends to be particularly voluminous. Compression enables therepresentation of large amounts of data using fewer bits, therebyincreasing storage capacity and reducing transmission times. Currenttechniques for video transmission include MPEG and its progeny, MPEG2and MPEG4. MPEG-type compression schemes divide the original image frameinto blocks or uniform size and shape, and transmit the motion, i.e.,the change in location of blocks from one frame to another. This reducesthe amount of data that needs to be transmitted and/or stored.

[0007] One relevant limitation of MPEG and other compression schemes isthat as blocks or objects move, new regions within the image may beuncovered. FIGS. 1A and 1B illustrate a newly uncovered (exposed) imageregion and are used to illustrate a problem that serves as a motivationfor the present invention. FIG. 1A illustrates an image frame composedof four regions or “objects,” marked 11 through 14 as illustrated. Eachobject may include multiple blocks under block-based compression schemessuch as MPEG. (The objects in FIG. 1 are rectangular for purposes ofsimplicity of explanation. Actual objects need not be, and typically arenot, of rectangular shape. In some schemes, the objects may be ofarbitrary shape.) In FIG. 1B, objects 12 and 14 have moved aparthorizontally, revealing image region 15, a previously occluded, nownewly uncovered region. The color values of region 15 are wholly unknownto the decoder. MPEG and similar programs simply apply one of many stillimage compression techniques, such as DCT coding, to the newly uncoveredregions and then transmits it to the receiving device. This conventionalway of dealing with newly uncovered regions is rather inefficient.

[0008] Multiscale Transforms

[0009] Examples of multi-scale transforms are found in the field ofimage and video processing. There applications include spectralanalysis, image de-noising, feature extraction, and, of course,image/video compression. JPEG2000, the Laplacian pyramid of Burt &Adelson [Burt and Adelson I], traditional convolution wavelet sub-banddecomposition, and the lifting implementation of [Sweldens I] are allexamples of multi-scale transforms. Many variations of multi-scaletransforms differ in regards to how the transform coefficients arequantized and then encoded. Such variations include SPIHT by Said andPearlman [SPIHTI], EZW (see [Shapiro I]), trellis coding (see [MarcellinI]), etc.

[0010] All multi-scale transforms operate on the principle that theefficient representation of a given multi-dimensional signal ischaracterized by looking at the data via a decomposition acrossdifferent scales. Here a scale refers to a characteristic length scaleor frequency. Coarse scales refer to smooth broad transitions in afunction. The very fine scales denote the often sharp, localfluctuations that occur at or near the fundamental pixel scale of thesignal.

[0011]FIG. 2A illustrates an example of different scale information fora given 1-D signal. Note that the function is actually wellcharacterized as a smoothly varying coarse scale function f1(x) (seeFIG. 2B) plus one other function depicted in FIG. 2C, f2(x). Thefunction f2(x) contains the majority of the fine scale information. Notethat f2(x) tends to oscillate or change on a very short spatial scale;whereas f1(x) changes slowly on a much longer spatial scale. Thecommunications analogy is that of a carrier signal (i.e. coarse scalemodulating signal) and the associated transmission band (i.e. highfrequency or fine scale signal). In fact by referring to FIGS. 2A-C onecan see that the complete high frequency details are well characterizedby f2(x) and the low frequency or average properties of the signal areexhibited by f1(x). In fact few signals are as cleanly characterizedinto specific scales as the function depicted in FIG. 2A.

[0012] FIGS. 2D-G show a similar process in 2-dimensions (2-D). Theoriginal pixel data, or finest scale, is denoted in FIG. 2D. Here theaveraging filter at each scale is depicted in FIG. 2E as well as anexample sub-sampling rule. In this case the sub-sampling rule isreferred to as a quincunx lattice in 2-dimensions and once againpreserves half the points at each step. FIGS. 2F and G show successivesteps in building the multi-resolution pyramid for a square domain viaapplication of the filter and sub-sampling logic depicted in FIG. 2E. Ateach step of the process the numbers at each pixel refer to thefunctional value of the pyramid at a given scale. Note that the scaledepicted in FIG. 2G contains almost one quarter of the sample points inthe original 2-D function shown in FIG. 2D because each application ofthe quincunx sub-sampling reduces the number of points by a factor oftwo. Other samplings are also known in the art.

[0013] In order to handle boundary effects for the convolution at theedge of the pictured rectangular domain, it may be assumed, for example,that the data at each scale can be extended via a mirror symmetricextension appropriate to the dimensionality of the signal across theboundary in question.

[0014] Pyramidal Transform

[0015]FIG. 2H depicts a conventional forward pyramidal transform 200.The transform 200 typically operates on an image 202. The pyramidaltransform 200 illustrated in FIG. 2H includes three levels (layers) oftransformation.

[0016] In the first level of transformation, a low pass reduction 204and a high pass reduction 206 are performed on the image 202. The lowpass reduction 204 comprises filtering the color values of the imagearray through a low pass filter, then reducing by down sampling. Forexample, if the down sampling is by a factor of two, then every otherpixel is effectively removed by the reduction. The result of such a lowpass reduction is a coarser version of the image. If the down samplingis by a factor of two, then the low pass reduction 204 outputs a coarsesubimage (not shown) with half the number of pixels as in the image 202.Similarly, the high pass reduction 206 comprises filtering the colorvalues of the image array through a high pass filter, then reducing bydown sampling. The result of such a high pass reduction is differencesubimage 208. The difference subimage 208 also has a fraction of thenumber of pixels as in the image 202. A difference subimage may becalled an error subimage. As described later, difference subimages maybe recombined with coarse subimages to reconstruct the image.

[0017] The second and third levels are similar to the first level. Inthe second level, a low pass reduction 210 and a high pass reduction 212are performed on the coarse subimage output by the first level's lowpass reduction 204. In the third level, a low pass reduction 216 and ahigh pass reduction 218 are performed on the coarse subimage output bythe second level's low pass reduction 210. The result of the pyramidaltransform 200 is a final coarse subimage 222 output by the third level'slow pass reduction and three difference subimages 208, 214, and 220 (onefor each level).

[0018]FIG. 2I depicts a conventional reverse (or inverse) pyramidaltransformation 250. The inverse transform 250 operates on the coarsesubimage 222 output by the forward transform 200. Like the forwardtransform 200 in FIG. 2H, the reverse transform 250 in FIG. 2I includesthree levels.

[0019] In the first level, an expansion low pass 252 is performed on thecoarse subimage 222. The expansion low pass 252 comprises expanding byupsampling, then filtering through a low pass filter. For example, ifthe up sampling is by a factor of two, then a zero pixel is effectivelyinserted between every two pixels. Also in the first level, expansionhigh pass 254 is performed on the last (in this case, the third)difference subimage 220 from the forward transform 200. The expansionhigh pass 254 comprises expanding by upsampling, then filtering througha high pass filter. The outputs of the expansion low pass 252 and of theexpansion high pass 254 are then added together. The result is a lesscoarse subimage (not shown). For example, if the upsampling is by afactor of two, then the less coarse subimage should have twice thenumber of pixels as the coarse subimage 222.

[0020] The second and third levels are similar to the first level. Inthe second level, an expansion low pass 256 is performed on the lesscoarse subimage output by the first level's expansion low pass 252. Inaddition, an expansion high pass 258 is performed on the seconddifference subimage 214 from the forward transform 200. The outputs ofthe expansion low pass 256 and of the expansion high pass 258 are thenadded together. The result is another less coarse subimage (not shown).In the third level, an expansion low pass 260 is performed on the lesscoarse subimage output by the second level's expansion low pass 256. Inaddition, an expansion high pass 262 is performed on the firstdifference subimage 208 from the forward transform 200. The outputs ofthe expansion low pass 260 and of the expansion high pass 262 are thenadded together. The result is a reconstruction of the image 202. Notethat the conventional transform and inverse transform as described aboveis lossless in that the reconstructed image 202 in FIG. 2I is the sameas the original image 202 in FIG. 2H.

SUMMARY

[0021] In accordance with an embodiment of the invention, a method ofencoding includes generating for each transform point a doubledifference coefficient (comprising the difference between a modeleddifference coefficient and a raw difference coefficient) and encoding asan adaptive difference coefficient for each transform point either thedouble difference coefficient or the raw difference coefficient. Whetherthe double difference coefficient or the raw difference coefficient isselected to be the adaptive difference coefficient depends on which oneprovides more efficient coding.

[0022] In accordance with an embodiment of the invention, a method ofdecoding includes receiving the adaptive difference coefficients fromthe encoder, applying the same modeling and transform as the encoder togenerate the modeled difference coefficients, generating correctivedifference coefficients (from the adaptive difference coefficients andthe modeled coefficients), and inverse transformation using thecorrective difference coefficients.

[0023] In accordance with an embodiment of the invention, a system mayinclude an encoder implementing the method of encoding and a decoderimplementing the method of decoding.

BRIEF DESCRIPTION OF THE DRAWINGS

[0024]FIGS. 1A and 1B illustrate a newly uncovered (exposed) imageregion.

[0025] FIGS. 2A-C illustrate a multiscale transform in one dimension.

[0026] FIGS. 2D-2G illustrate multiscale transforms in two dimensions.

[0027]FIG. 2H depicts a conventional forward pyramidal transform.

[0028]FIG. 2I depicts a conventional reverse pyramidal transformation.

[0029]FIG. 3 depicts a forward transform of an image region with rawdata in accordance with an embodiment of the invention.

[0030]FIG. 4 depicts a forward transform of an image region with modeleddata in accordance with an embodiment of the invention.

[0031]FIG. 5 depicts generation of double difference coefficients inaccordance with an embodiment of the invention.

[0032]FIG. 6 is a flow chart depicting encoding of adaptive differencecoefficients in accordance with an embodiment of the invention.

[0033]FIG. 7 is a flow chart depicting determination of correctivedifference coefficients in accordance with an embodiment of theinvention.

[0034]FIG. 8 depicts a reverse transform to decode an image region inaccordance with an embodiment of the invention.

[0035]FIG. 9 is a flow chart depicting an encoding process in accordancewith an embodiment of the invention.

[0036]FIG. 10 is a flow chart depicting a decoding process in accordancewith an embodiment of the invention.

DESCRIPTION OF THE SPECIFIC EMBODIMENTS

[0037]FIG. 3 depicts a forward transform of an image region with rawdata in accordance with an embodiment of the invention. The transform300 may operate on a region of an image. For example, the region maycomprise an image block or an image segment.

[0038] The specific forward transform 300 in FIG. 3 includes eightlevels of transformation. If the down sampling in each level is by afactor of two, then having eight levels results in a reduction in pixelsby a factor of 256 (2⁸). Square regions of 256 (16×16) pixels or lessmay be reduced to a single average pixel by such an eight-leveltransform 300. In one specific embodiment, the regions are such thatthey generally fit within such a 16×16 block, so that the result of theeight-level transform 300 is a single average pixel. However, transformswith other numbers of levels of transformation are also contemplated tobe within the scope of this invention. For example, a four-leveltransform may reduce regions that fit within an 4×4 square into a singleaverage pixel, a six-level transform may reduce regions that fit withinan 8×8 square into a single average pixel, a ten-level transform mayreduce regions that fit within an 32×32 square into a single averagepixel, a twelve-level transform may reduce regions that fit within an64×64 square into a single average pixel, a fourteen-level transform mayreduce regions that fit within an 128×128 square into a single averagepixel, a sixteen-level transform may reduce regions that fit within an256×256 square into a single average pixel, and so on.

[0039] In the first level of transformation, a low pass reduction 302and a high pass reduction 303 are performed on the image region with rawdata 301. The low pass reduction 302 comprises filtering the raw colorvalues of the image region 301 through a low pass filter, then reducingby down sampling. For example, if the down sampling is by a factor oftwo, then every other pixel is effectively removed by the reduction. Theresult of such a low pass reduction is a coarser version of the rawimage region. If the down sampling is by a factor of two, then the lowpass reduction 302 outputs a coarse raw subregion (not shown) with halfthe number of pixels as in the raw region 301. Similarly, the high passreduction 303 comprises filtering the raw region 301 through a high passfilter, then reducing by down sampling. The result of such a high passreduction is a first set of raw differences (i.e. a raw differencesubregion) 304. The raw difference subregion 304 also has a fraction ofthe number of pixels as in the image region 301.

[0040] The second through eighth levels are similar to the first level.In the each level, a low pass reduction 305, 308, . . . , 323 and a highpass reduction 306, 309, . . . , 324 are performed on the coarse rawsubregion output by the prior level's low pass reduction. If the imageregion 301 is small enough (i.e. fits within a 16×16 block for an8-level transform), then the final coarse raw subregion 326 output bythe last level's low pass reduction should comprise a single pixel thatrepresents the average pixel of the raw region 301. In such instances,the result of the forward transform 300 is the average raw pixel of theregion 326 and eight raw difference subregions 304, 307, 310, . . . ,325 (one for each level). If the image region 301 is small enough suchthat it can be processed by less than the eight levels, then theprocessing may end before the last levels. For example, if the imageregion 301 fits within a 8×8 block, only six levels are needed forprocessing.

[0041]FIG. 4 depicts a forward transform of an image region with modeleddata in accordance with an embodiment of the invention. The forwardtransform 400 of FIG. 4 transforms the region with modeled data 401 inthe same way that the forward transform 300 of FIG. 3 transforms theregion with raw data 301. The result of the forward transform 400 ofFIG. 4 is the average modeled pixel of the region 426 and eight modeleddifference subregions 404, 407, 410, . . . , 425 (one for each level).

[0042]FIG. 5 depicts generation of double difference coefficients inaccordance with an embodiment of the invention. As shown in FIG. 5, thedouble difference coefficients are generated by taking the modeleddifference coefficients and subtracting the raw difference coefficients.This is done per transform point. The generation process 500 depicted inFIG. 5 results in eight sets of double difference coefficients 504, 507,510, . . . , 525. The first set of double difference coefficients 504 isgenerated by subtracting (on a point by point basis) the first set ofraw difference coefficients 304 from the first set of modeled differencecoefficients 404. And so on to generate the other seven sets of doubledifference coefficients 507, 510, . . . , 525.

[0043]FIG. 6 is a flow chart depicting encoding of adaptive differencecoefficients in accordance with an embodiment of the invention. Thisprocess 600 is generally performed in an encoder. The specific process600 depicted encodes a single adaptive difference coefficient. Hence,the process 600 is to be applied to each transform point to encode allthe adaptive difference coefficients. According to the process 600, adetermination is made as to whether the double difference coefficient orthe raw difference coefficient is more efficient to encode (step 602).If the double difference coefficient is more efficient to encode, thenthe double difference coefficient is encoded as the adaptive differencecoefficient for the transform point (step 604). On the other hand, ifthe raw difference coefficient is more efficient to encode, then the rawdifference coefficient is encoded as the adaptive difference coefficientfor the transform point (step 606).

[0044] Generally, the smaller the coefficient, the more efficient it isto encode. If the modeled data is close to the raw data for the relevantpixel area, then the double difference coefficient would tend to berelatively small and efficient to encode. However, if the modeled datais quite different from the raw data for the relevant pixel area, thenthe double difference coefficient would tend to be relatively large andinefficient to encode. In that case, the raw difference coefficient maybe smaller and more efficient to encode.

[0045]FIG. 7 is a flow chart depicting determination of correctivedifference coefficients in accordance with an embodiment of theinvention. This process 700 is generally performed in a decoder. Thecorrective difference coefficient at each point depends on the type ofadaptive difference coefficient encoded at that point (step 702). If theadaptive difference coefficient at a transform point is a doubledifference coefficient, then the corrective difference coefficient isset to be the modeled difference coefficient for that point minus thedouble difference coefficient for that point (step 704). On the otherhand, if the adaptive difference coefficient at a transform point is araw difference coefficient, then the corrective difference coefficientis simply set to be the raw difference coefficient for that point (step706). In accordance with one embodiment, one flag per transform point(with entropy coding we would expect much higher coding efficiency than1 bit point) may be transmitted by the encoder to indicate whether theadaptive difference coefficient is the double difference coefficient orthe raw difference coefficient. Alternatively, the decoder may possiblybe able to determine deductively which one it is.

[0046]FIG. 8 depicts a reverse transform to decode an image region inaccordance with an embodiment of the invention. The reverse transform800 may begin its operates on the average modeled pixel (or perhaps verycoarse subregion) 426 output by the forward transform 400. Like theforward transform 400 in FIG. 4, the reverse transform 800 of FIG. 8includes eight levels. In other words, the same number of levels areused in the reverse transform as were used in the forward transform.

[0047] In the first level, an expansion low pass 802 is performed on theaverage modeled pixel (or perhaps very coarse subregion) 426. Theexpansion low pass 802 comprises expanding by upsampling, then filteringthrough a low pass filter. For example, if the up sampling is by afactor of two, then a zero pixel is effectively inserted between everytwo pixels. Also in the first level, expansion high pass 806 isperformed on the last (in this case, the eighth) set of correctivedifference coefficients 804. The expansion high pass 806 comprisesexpanding by upsampling, then filtering through a high pass filter. Theoutputs of the expansion low pass 802 and of the expansion high pass 806are then added together. The result is a less coarse subregion (notshown). For example, if the upsampling is by a factor of two, then theless coarse subregion should have twice the number of pixels as thatinput into the first level.

[0048] The second and third levels are similar to the first level. Inthe second level, an expansion low pass 808 is performed on the lesscoarse subregion output by the first level's expansion low pass 802. Inaddition, an expansion high pass 812 is performed on the second set ofcorrective differences 810. The outputs of the expansion low pass 808and of the expansion high pass 812 are then added together. The resultis another less coarse subimage (not shown). And so on for the thirdthrough eighth levels. The result of the decoding process 800 is adecoded image region 850 that should be a good approximation to the rawimage region 301.

[0049]FIG. 9 is a flow chart depicting an encoding process in accordancewith an embodiment of the invention. This process 900 is generallyperformed in an encoder. The process 900 includes a step 902 wheremodeling (prediction) is performed for the image region at issue. Any ofvarious types of modeling may be used. For purposes of illustration, ifthe region at issue is a newly uncovered (exposed) region such as region15 in FIG. 1, then that region 15 may be modeled by selecting one of thesurrounding segments 11-14 and extending the pixel values from theselected segment into the region 15. For example, neighboring segment 12may be selected and extrapolated to model the pixel data of region 15.That is merely an example of one type of modeling that may be used.Various other types of modeling (prediction) may be used within thescope of the invention.

[0050] Once the region is modeled 902, the modeled image region 401 istransformed 400 in accordance with the forward transform 400 describedin relation to FIG. 4. In addition, the raw image region 301 istransformed in accordance with the forward transform 300 described inrelation to FIG. 3. Note that the encoder is able to the latter forwardtransform 300 because it has the raw data of the image region. Incontrast, the decoder does not have the raw data.

[0051] Next, the double difference coefficients are determined 500 inaccordance with FIG. 5. This step 500 requires use of the outputs ofboth the forward transforms 300 and 400. Using the double differencecoefficients and the raw difference coefficients, an adaptive encoding600 is performed in accordance with FIG. 6. The adaptive encodingselects at each transform point either the double difference coefficientor the raw difference coefficient. The more efficient of the twocoefficients is the one selected to be encoded as the adaptivedifference coefficient for that transform point. Finally, the adaptivedifference coefficients are transmitted 904 by the encoder to thedecoder.

[0052]FIG. 10 is a flow chart depicting a decoding process in accordancewith an embodiment of the invention. The process 1000 is generallyperformed in a decoder. The process 1000 includes a step 902 wheremodeling (prediction) is performed for the image region at issue. Themodeling 902 performed in the decoding process 1000 is the same modeling1002 performed by the encoding process 900. This feature is utilized bythe present invention to decrease the bandwidth needed to transmit anencoded image region. Because the decoder does the modeling 902, it doesnot need to receive the modeled difference coefficients. However, itdoes receive the adaptive difference coefficients transmitted by theencoder (step 904).

[0053] The modeled data of the region are forward transformed 400 inaccordance with FIG. 4. The forward transform 400 used by the decoder isthe same as the one used by the encoder. Using the modeled differencecoefficients from the transform 400 and the adaptive differencecoefficients from the encoder, the decoder determines the correctivedifference coefficients in accordance with FIG. 7. Finally, using thecorrective difference coefficients (and the average modeled pixel), thedecoder performs an inverse transform in accordance with FIG. 8. Theresult of which is the decoded image region 850 that should be a goodapproximation to the raw image region 301.

[0054] Note that embodiments of the present invention may be applied toblock-based compression schemes, such as MPEG-like schemes, and also tosegment-based compression schemes, such as that described in theabove-referenced “Segmentation” and “Video Processing” applications.

[0055] For example, referring back to FIG. 1B, in many real-lifesituations, the color of region 15 is related to the color values of one(or possible more than one) of the surrounding segments. Therefore usingthe surrounding segments, a smart decoder may predict (i.e. model) thelikely color values of the region 15. (As used herein, an image segmentmay be equivalent to an image region.) The above description provides anarchitecture for an exemplary encoder/decoder system that efficientlytransmits information sufficient for the (smart) decoder to decode withreasonable accuracy the color values of such newly uncovered (exposed)image regions. The encoder and decoder are synchronized in that theencoder and decoder use the same prediction (modeling) algorithms sothat only correction-related information needs to be sent to thedecoder. Referring back to FIG. 1B, the encoder and decoder, knowing thesurrounding regions 11-14, will make identical predictions as to thecolor values of the newly exposed image region 15. The encoder will thenonly need to transmit the correction-related information to the decoder.This method is efficient because the correction-related information istypically relatively small or may be made to be small by theabove-described adaptive double pyramidal coding.

[0056] In the above description, numerous specific details are given toprovide a thorough understanding of embodiments of the invention.However, the above description of illustrated embodiments of theinvention is not intended to be exhaustive or to limit the invention tothe precise forms disclosed. One skilled in the relevant art willrecognize that the invention can be practiced without one or more of thespecific details, or with other methods, components, etc. In otherinstances, well-known structures or operations are not shown ordescribed in detail to avoid obscuring aspects of the invention. Whilespecific embodiments of, and examples for, the invention are describedherein for illustrative purposes, various equivalent modifications arepossible within the scope of the invention, as those skilled in therelevant art will recognize.

[0057] These modifications can be made to the invention in light of theabove detailed description. The terms used in the following claimsshould not be construed to limit the invention to the specificembodiments disclosed in the specification and the claims. Rather, thescope of the invention is to be determined by the following claims,which are to be construed in accordance with established doctrines ofclaim interpretation.

What is claimed is:
 1. A method for encoding an image region using atransform, the method comprising: generating for each transform point adouble difference coefficient, wherein the double difference coefficientcomprises a difference between a raw difference coefficient and amodeled difference coefficient; and encoding as an adaptive differencecoefficient for each transform point either the double differencecoefficient or the raw difference coefficient, wherein whether thedouble difference coefficient or the raw difference coefficient isselected to be the adaptive difference coefficient depends on which oneprovides more efficient coding.
 2. The method of claim 1 wherein imageregion comprises an exposed region of an image.
 3. The method of claim 1wherein the transform comprises a pyramidal transform.
 4. The method ofclaim 2 wherein the pyramidal transform includes more than three levelsof transformation.
 5. A method for decoding an image region using atransform, the method comprising: receiving adaptive differencecoefficients from an encoder; modeling the image region to generate amodeled region; transforming the modeled region to generate modeleddifference coefficients; determining corrective difference coefficientsusing the adaptive difference coefficients and the modeled differencecoefficients; and inverse transformation using the corrective differencecoefficients.
 6. The method of claim 5 wherein the adaptive differencecoefficients comprise double difference coefficients and raw differencecoefficients.
 7. The method of claim 6 wherein, if the adaptivedifference coefficient at a transform point is the raw coefficient, thenthe corrective difference coefficient at the transform point comprisesthe raw difference coefficient.
 8. The method of claim 7 wherein, if theadaptive difference coefficient at a transform point is the doubledifference coefficient, then the corrective difference coefficient atthe transform point comprises the modeled difference coefficient minusthe double difference coefficient.
 9. The method of claim 5 wherein thetransform comprises a pyramidal transform.
 10. The method of claim 9wherein the pyramidal transform includes more than three levels oftransformation.
 11. An encoder for encoding an image region using atransform, the encoder comprising: means for generating for eachtransform point a double difference coefficient, wherein the doubledifference coefficient comprises a difference between a raw differencecoefficient and a modeled difference coefficient; and means for encodingas an adaptive difference coefficient for each transform point eitherthe double difference coefficient or the raw difference coefficient,wherein whether the double difference coefficient or the raw differencecoefficient is selected to be the adaptive difference coefficientdepends on which one provides more efficient coding.
 12. A decoder fordecoding an image region using a transform, the decoder comprising:means for receiving adaptive difference coefficients from an encoder;means for modeling an image to generate a modeled region; means fortransforming the modeled region to generate modeled differencecoefficients; means for determining corrective difference coefficientsusing the adaptive difference coefficients and the modeled differencecoefficients; and means for inverse transformation using the correctivedifference coefficients.
 13. A system for decoding and encoding an imageregion using a transform, the system comprising: an encoder including(a) means for generating for each transform point a double differencecoefficient, wherein the double difference coefficient comprises adifference between a raw difference coefficient and a modeled differencecoefficient, and (b) means for encoding as an adaptive differencecoefficient for each transform point either the double differencecoefficient or the raw difference coefficient, wherein whether thedouble difference coefficient or the raw difference coefficient isselected to be the adaptive difference coefficient depends on which oneprovides more efficient coding; and a decoder including (a) means forreceiving the adaptive difference coefficients from the encoder, (b)means for modeling the image region to generate a modeled region, (c)means for transforming the modeled region to generate modeled differencecoefficients, (d) means for determining corrective differencecoefficients using the adaptive difference coefficients and the modeleddifference coefficients, and (e) means for inverse transformation usingthe corrective difference coefficients.